John Parkes

By Vyacheslav O. Vakhnenko, E. John Parkes and Dmitri B. Vengrovich

The physical phenomena that take place in nature generally have complicated nonlinear features. A variety of methods for examining the properties and solutions of nonlinear evolution equations are explored by using the Vakhnenko equation (VE) as an example. One remarkable feature of the VE is that it possesses loop-like soliton solutions. Loop-like solitons are a class of interesting wave phenomena, which have been involved in some nonlinear systems. The VE can be written in an alternative form, known as the Vakhnenko-Parkes equation (VPE). The VPE can be written in Hirota bilinear form. The Hirota method not only gives the N-soliton solution but enables one to find a way from the Bäcklund transformation through the conservation laws and associated eigenvalue problem to the inverse scattering transform (IST) method. This method is the most appropriate way of tackling the initial value problem (Cauchy problem). The standard procedure for IST method is expanded for the case of multiple poles, specifically, for the double poles with a single pole. In recent papers some physical phenomena in optics and magnetism are satisfactorily described by means of the VE. The question of physical interpretation of multivalued (loop-like) solutions is still an open question.

Part of the book: Research Advances in Chaos Theory